To find the zeros of the polynomial function \( f(x) = x^3 - 5x^2 + 6x - 30 \), we can try to find potential rational zeros using the Rational Root Theorem, which suggests that any rational root, in the form \( \frac{p}{q} \), can be found where \( p \) divides the constant term (-30), and \( q \) divides the leading coefficient (1).
The possible rational roots are \( \pm 1, \pm 2, \pm 3, \pm 5, \pm 6, \pm 10, \pm 15, \pm 30 \).
Let's try testing some of these possible rational roots in the polynomial.
- Testing \( x = 2 \):
\[ f(2) = 2^3 - 5(2^2) + 6(2) - 30 = 8 - 20 + 12 - 30 = -30 \quad (\text{not a root}) \]
- Testing \( x = 3 \):
\[ f(3) = 3^3 - 5(3^2) + 6(3) - 30 = 27 - 45 + 18 - 30 = -30 \quad (\text{not a root}) \]
- Testing \( x = 5 \):
\[ f(5) = 5^3 - 5(5^2) + 6(5) - 30 = 125 - 125 + 30 - 30 = 0 \quad (\text{this is a root}) \]
Now that we found \( x = 5 \) is a root, we can use synthetic division to divide \( f(x) \) by \( x - 5 \).
Synthetic Division
Set up the synthetic division with the coefficients of \( f(x) \):
- Coefficients are: \( 1, -5, 6, -30 \)
Here is how we set up the synthetic division using \( x = 5 \):
\[ \begin{array}{r|rrrr} 5 & 1 & -5 & 6 & -30 \ & & 5 & 0 & 30 \ \hline & 1 & 0 & 6 & 0 \ \end{array} \]
Explanation of the steps:
- Bring down the leading coefficient (1).
- Multiply this number by the root (5) and put it in the next column (-5), giving us \( -5 + 5 = 0 \).
- Multiply the result (0) by 5 and place it under the next coefficient (6), resulting in \( 6 + 0 = 6 \).
- Multiply the result (6) by 5 and place it under -30, resulting in \( -30 + 30 = 0 \).
The final row \( 1, 0, 6, 0 \) represents the polynomial \( x^2 + 0x + 6 \) or simply \( x^2 + 6 \).
Now we can set the remaining polynomial \( x^2 + 6 \) to zero to find the other roots:
\[ x^2 + 6 = 0 \quad \Rightarrow \quad x^2 = -6 \quad \Rightarrow \quad x = \pm i\sqrt{6} \]
Final Roots
So, the complete set of zeros for the polynomial \( f(x) = x^3 - 5x^2 + 6x - 30 \) is:
\[ x = 5, \quad x = i\sqrt{6}, \quad x = -i\sqrt{6} \]